## Journal of Symbolic Logic

### Hybrid Logics: Characterization, Interpolation and Complexity

#### Abstract

Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called $\mathscr{H}(\downarrow, @)$. We show in detail that $\mathscr{H}(\downarrow, @)$ is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Fraisse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that $\mathscr{H}(\downarrow, @)$ corresponds to the fragment of first-order logic which is invariant for generated submodels. We then show that $\mathscr{H}(\downarrow, @)$ enjoys (strong) interpolation, provide counterexamples for its finite variable fragments, and show that weak interpolation holds for the sublanguage $\mathscr{H}$(@). Finally, we provide complexity results for $\mathscr{H}$(@) and other fragments and variants, and sharpen known undecidability results for $\mathscr{H}(\downarrow, @)$.

#### Article information

Source
J. Symbolic Logic, Volume 66, Issue 3 (2001), 977-1010.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183746543

Mathematical Reviews number (MathSciNet)
MR1856725

Zentralblatt MATH identifier
0984.03018

JSTOR